My make-belief etymology is unintentionally adding sexual ambiguity this dialog. (In the Napoleonic/Regency AO fic “agony” or “private agony” is used instead of “heat’)
But, back to canon:
Charles has vocabuary quirks: “Groovy” as period appropriate aside, there’s “access your memory system” (this is giving me headcanon regarding conversations with Howard Stark, btw), “gotten” in British accent, but here he chooses “agony” in speaking with Erik. Charles’ been waiting for Erik to come out of the facility, there’s no reason not to believe he rehearsed or at least envisioned their conversation.
Agony is associated with pain, but more specifically, agony is pain/effort from competition (Greek root: “agon”), agony is also a word that denote the conflict between higher and lower orders of human nature in Christian theology. Even in the modern usage, the idea of a struggle has remain shaded into its meaning, giving it a more metaphysical weight than “pain”. Charles could’ve said “pain”. In fact, he does say “pain and anger” when he talks to Erik later in the film for a different purpose, but in this instance, he uses the word “agony” indicative of not only the raw experience of Erik’s past, the “Everything” that Erik thought should prompt Charles to stay away from him, but also of Erik’s internal self, his soul/mind/will that assimilates those experiences, translating them to decision and action. “I’ve felt your agony” is shorthand for “I know how you question yourself afterwards, I know the effort it requires to live, to go on…” Charles acknowledgement of Erik’s existential conflicts is a beautiful contrast to how sleek and determined 007 Nazi Killer!Erik has so far appears on screen. Charles shatters appearances. It’s a pivotal moment in their relationship. He offers help and Erik, self-efficient Erik, hesitates. He accepts.
It is even possible that in seeing what “Shaw did” to Erik includes the warping of Erik’s self-perception, his belief in his own goodness which Charles fights to the extent for even allowing Erik to leave him paralyzed on a beach surrounded by hostiles (and covered in the other meta of Charles vs. Shaw).
Of course, it’s almost 4am and I’m sure I’m reading unconscious decisions by the script writers who probably just want Charles to use a curious word and avoid memories of Clinton.